You can find some amusing (I might say amazing) papers in this area by searching for "primes at a glance" and " primes at a (somewhat lengthy) glance".
In the first paper they give (along with many other interesting things) what they "believe to be a complete list" of all pairs of integers $B,L$ with
- $N=B+L$
- $B \ge |L|$
- $\gcd(B,L)=1$
- if $p \le\sqrt{N}$, then $p$ divides $BL.$
- if $Q | BL$ and $p \lt q$ then $p | BL$
For $N \gt 1$ this proves $N$ to be prime as it rules out any proper divisors. Such a presentation provides an at a glance proof that $p$ is prime. For $N=31$ the presentations range from
$31=2^4+3\cdot5$
to
$31=2^3\cdot7\cdot11\cdot17\cdot23-3\cdot5^2\cdot13^2\cdot19$
The first primes for which they give no solutions are $541,547$
the last for which they do is
$2521=19\cdot43\cdot37\cdot2\cdot3^2\cdot5\cdot29^2\cdot41\cdot47^2-7\cdot11^3\cdot13\cdot17^2\cdot23^2\cdot31$
Without condition 5 there are many solutions. $88711$ is the product of $7,19,23,29$ and $72930$ is the product of $2,3,5,11,13,17$ so we can certainly find positive coprime integers with $1=88711x-72930y.$ Then $31=88711s-72930t$ is a difference of coprime values for $s=31x+72930$ and $t=31y+88711$ You can always do that.But probably not with $st$ having all prime factors below 31 ( in which case the prime factors would split the same way, given that none of them divide $31$.)